You’ve operated with fractions and decimals, prefer 3.8 and

*
. This numbers can be found between the integer numbers on a number line. Over there are various other numbers that can be uncovered on a number line, too. As soon as you include all the number that can be placed on a number line, you have the actual number line. Let"s destruction deeper into the number line and see what those numbers look like. Let’s take a closer look at to watch where this numbers fall on the number line.

You are watching: Name all sets to which each number belongs


The portion , mixed number

*
, and decimal 5.33… (or ) all represent the same number. This number belongs come a set of numbers that mathematicians contact rational numbers. Reasonable numbers space numbers that can be composed as a proportion of two integers. Nevertheless of the form used,  is rational since this number can be composed as the ratio of 16 end 3, or .

Examples of rational numbers incorporate the following.

0.5, as it can be composed as

*
, as it deserve to be written as
*

−1.6, as it deserve to be composed as

*

4, together it have the right to be written as

*

-10, as it deserve to be written as

*

All of these numbers can be written as the ratio of 2 integers.

You can locate these points ~ above the number line.

In the following illustration, points are displayed for 0.5 or , and also for 2.75 or

*
.

*

As you have actually seen, rational numbers deserve to be negative. Each hopeful rational number has actually an opposite. Opposing of  is

*
, for example.

Be cautious when put negative numbers ~ above a number line. The an unfavorable sign means the number is come the left that 0, and also the absolute worth of the number is the distance from 0. For this reason to place −1.6 on a number line, you would uncover a suggest that is |−1.6| or 1.6 systems to the left of 0. This is much more than 1 unit away, but less 보다 2.

*


Example

Problem

Place

*
 on a number line.

It"s helpful to an initial write this improper fraction as a combined number: 23 divided by 5 is 4 v a remainder of 3, so

*
 is .

Since the number is negative, you can think of it as relocating

*
 units come the left that 0.  will be between −4 and also −5.

Answer

*


Which the the following points to represent ?

*


Show/Hide Answer

A)

Incorrect. This allude is simply over 2 units to the left of 0. The point should it is in 1.25 devices to the left that 0. The correct answer is suggest B.

B)

Correct. An adverse numbers are to the left the 0, and  should be 1.25 units to the left. Allude B is the only suggest that’s an ext than 1 unit and less than 2 systems to the left that 0.

C)

Incorrect. An alert that this point is between 0 and also the very first unit note to the left the 0, for this reason it represents a number in between −1 and 0. The point for  should it is in 1.25 units to the left of 0. You may have correctly uncovered 1 unit to the left, however instead of proceeding to the left another 0.25 unit, you moved right. The correct answer is allude B.

D)

Incorrect. Negative numbers are to the left of 0, no to the right. The suggest for  should it is in 1.25 devices to the left that 0. The exactly answer is point B.

E)

Incorrect. This point is 1.25 units to appropriate of 0, for this reason it has actually the correct distance yet in the not correct direction. An adverse numbers are to the left of 0. The correct answer is allude B.

Comparing rational Numbers


When 2 whole numbers room graphed top top a number line, the number to the appropriate on the number line is constantly greater 보다 the number top top the left.

The exact same is true when comparing 2 integers or rational numbers. The number come the ideal on the number line is always greater than the one ~ above the left.

Here space some examples.


Numbers come Compare

Comparison

Symbolic Expression

−2 and also −3

−2 is better than −3 because −2 is to the ideal of −3

−2 > −3 or −3 −2

2 and 3

3 is higher than 2 due to the fact that 3 is to the appropriate of 2

3 > 2 or 2

−3.5 and also −3.1

−3.1 is greater than −3.5 since −3.1 is to the right of −3.5 (see below)

−3.1 > −3.5 or

−3.5 −3.1


*

Which the the complying with are true?

i. −4.1 > 3.2

ii. −3.2 > −4.1

iii. 3.2 > 4.1

iv. −4.6

A) i and iv

B) i and ii

C) ii and iii

D) ii and also iv

E) i, ii, and also iii


Show/Hide Answer

A) i and iv

Incorrect. −4.6 is come the left of −4.1, for this reason −4.6 −4.1 or −4.1 −4.1 and also −4.6

B) i and also ii

Incorrect. −3.2 is come the appropriate of −4.1, so −3.2 > −4.1. However, positive numbers such as 3.2 are constantly to the right of an adverse numbers such together −4.1, so 3.2 > −4.1 or −4.1 ii and also iv, −3.2 > −4.1 and also −4.6

C) ii and iii

Incorrect. −3.2 is to the best of −4.1, so −3.2 > −4.1. However, 3.2 is to the left of 4.1, for this reason 3.2 ii and also iv, −3.2 > −4.1 and −4.6

D) ii and iv

Correct. −3.2 is come the best of −4.1, therefore −3.2 > −4.1. Also, −4.6 is to the left of −4.1, so −4.6

E) i, ii, and also iii

Incorrect. −3.2 is to the best of −4.1, for this reason −3.2 > −4.1. However, positive numbers such together 3.2 are constantly to the appropriate of an adverse numbers such together −4.1, so 3.2 > −4.1 or −4.1 ii and iv, −3.2 > −4.1 and also −4.6


Irrational and also Real Numbers


There are also numbers that space not rational. Irrational numbers cannot be composed as the proportion of 2 integers.

Any square root of a number that is not a perfect square, for instance , is irrational. Irrational numbers space most frequently written in among three ways: as a source (such together a square root), using a unique symbol (such together ), or as a nonrepeating, nonterminating decimal.

Numbers with a decimal part can either be terminating decimals or nonterminating decimals. Terminating method the number stop at some point (although you can always write 0s at the end). Because that example, 1.3 is terminating, because there’s a critical digit. The decimal type of  is 0.25. Terminating decimals are always rational.

Nonterminating decimals have actually digits (other than 0) that proceed forever. Because that example, take into consideration the decimal form of

*
, which is 0.3333…. The 3s continue indefinitely. Or the decimal kind of
*
 , i m sorry is 0.090909…: the succession “09” proceeds forever.

In addition to being nonterminating, these two numbers are additionally repeating decimals. Your decimal parts are make of a number or sequence of numbers that repeats again and also again. A nonrepeating decimal has digits that never form a repeating pattern. The worth of, for example, is 1.414213562…. No issue how much you lug out the numbers, the number will never ever repeat a ahead sequence.

If a number is terminating or repeating, it must be rational; if the is both nonterminating and nonrepeating, the number is irrational.


Type of Decimal

Rational or Irrational

Examples

Terminating

Rational

0.25 (or )

1.3 (or

*
)

Nonterminating and Repeating

Rational

0.66… (or

*
)

3.242424… (or)

*

Nonterminating and also Nonrepeating

Irrational

 (or 3.14159…)

*
(or 2.6457…)


*


Example

Problem

Is 82.91 reasonable or irrational?

Answer

−82.91 is rational.

The number is rational, since it is a terminating decimal.


The set that real numbers is do by combine the collection of reasonable numbers and also the set of irrational numbers. The actual numbers encompass natural numbers or counting numbers, entirety numbers, integers, rational number (fractions and also repeating or end decimals), and also irrational numbers. The set of actual numbers is all the number that have a place on the number line.

Sets that Numbers

Natural number 1, 2, 3, …

Whole number 0, 1, 2, 3, …

Integers …, −3, −2, −1, 0, 1, 2, 3, …

Rational number numbers that deserve to be composed as a proportion of 2 integers—rational numbers are terminating or repeating once written in decimal form

Irrational number numbers 보다 cannot be composed as a proportion of two integers—irrational numbers are nonterminating and also nonrepeating once written in decimal form

Real numbers any number that is reasonable or irrational


Example

Problem

What set of number does 32 belong to?

Answer

The number 32 belong to all these to adjust of numbers:

Natural numbers

Whole numbers

Integers

Rational numbers

Real numbers

Every herbal or counting number belong to every one of these sets!


Example

Problem

What sets of number does

*
 belong to?

Answer

 belongs to these sets that numbers:

Rational numbers

Real numbers

The number is rational because it"s a repeating decimal. It"s equal to

*
 or
*
 or .


Example

Problem

What to adjust of number does

*
 belong to?

Answer

*
 belongs to these sets of numbers:

Irrational numbers

Real numbers

The number is irrational since it can"t be written as a proportion of two integers. Square roots that aren"t perfect squares are constantly irrational.


Which the the following sets go

*
 belong to?

whole numbers

integers

rational numbers

irrational numbers

real numbers

A) rational number only

B) irrational number only

C) rational and real numbers

D) irrational and also real numbers

E) integers, rational numbers, and real numbers

F) entirety numbers, integers, reasonable numbers, and real numbers


Show/Hide Answer

A) reasonable numbers only

Incorrect. The number is rational (it"s created as a ratio of 2 integers) but it"s additionally real. Every rational number are likewise real numbers. The correct answer is rational and also real numbers, since all rational numbers are additionally real.

B) irrational numbers just

Incorrect. Irrational number can"t be created as a ratio of two integers. The correct answer is rational and also real numbers, due to the fact that all rational numbers are likewise real.

C) rational and also real number

Correct. The number is between integers, so that can"t be an integer or a entirety number. It"s composed as a ratio of two integers, therefore it"s a reasonable number and not irrational. Every rational number are genuine numbers, therefore this number is rational and real.

D) irrational and also real number

Incorrect. Irrational number can"t be composed as a ratio of 2 integers. The correct answer is rational and real numbers, because all rational numbers are likewise real.

E) integers, rational numbers, and also real number

Incorrect. The number is in between integers, no an creature itself. The exactly answer is rational and real numbers.

F) totality numbers, integers, reasonable numbers, and real numbers

Incorrect. The number is between integers, so it can"t it is in an integer or a totality number. The exactly answer is rational and real numbers.

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Summary


The set of real numbers is every numbers that can be displayed on a number line. This contains natural or count numbers, totality numbers, and integers. It likewise includes rational numbers, which are numbers that can be created as a ratio of 2 integers, and also irrational numbers, which cannot be composed as a the proportion of two integers. When comparing two numbers, the one with the better value would appear on the number heat to the ideal of the other one.